In recent years, a tight connection has emerged between modal logic on
the one hand and coalgebras, understood as generic transition systems,
on the other hand. Here, we prove that (finitary) coalgebraic modal
logic has the finite model property. This fact not only reproves known
completeness results for coalgebraic modal logic, which we push further
by establishing that every coalgebraic modal logic admits a complete
axiomatization of rank 1; it also enables us to establish a generic
decidability result and a first complexity bound. Examples covered by
these general results include, besides standard Hennessy-Milner logic,
graded modal logic and probabilistic modal logic.